This form of model involves direct minimal curvature matter coupling, which has been used widely to explore
a number of cosmological phenomena because of its theoretical and cosmological consistency [1]}.
Many theoretical cosmological models of the universe have been proposed to analyze the behavior of
mysterious components and their physical and cosmological consequences are explored [2]}, [3]}.
Galactic structures and their existence have also been discussed, and results are found in agreement with
previously established solutions and assumptions [4]}.
For this model, homologous condition and homogeneous expansion condition given in Eqs.\((\ref {105})\)  and \((\ref {106i})\)  take the form as
\((1+\alpha _2)qG &=&\sqrt{3}\frac{\sigma H^{\prime }}{H},\\\left(\sqrt{3} \frac{\sigma }{H}+\frac{1}{3}D_H\sigma \right)&=& -\frac{qG}{H^{\prime }}(1+\alpha _2).\) 
